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RMVT- and PVD-based nite cylindrical layer methodsfor the three-dimensional buckling analysis of multilayeredFGM cylinders under axial compression

Chih-Ping Wu , Shu-Ting Peng, Yen-Cheng ChenDepartment of Civil Engineering, National Cheng Kung University, Tainan 70101, Taiwan, ROC

a r t i c l e i n f o

Article history:Received 21 September 2012Received in revised form 2 May 2013Accepted 24 June 2013Available online 4 July 2013

Keywords:Finite layer methodsVariational principlesBucklingFunctionally graded materialsLaminatesCylinders

a b s t r a c t

The unied formulations of nite cylindrical layer methods (FCLMs) based on the Reissnermixed variational theorem (RMVT) and the principle of virtual displacements (PVD) aredeveloped for the three-dimensional (3D) linear buckling analysis of simply-supported,multilayered functionally graded material (FGM) circular hollow cylinders and laminatedcomposite ones under axial compression. The material properties of the FGM layer areassumed to obey the power-law distributions of the volume fraction of the constituentsthrough the thickness coordinate. In these formulations, the cylinder is divided into a num-ber of nite cylindrical layers, in which the trigonometric functions and Lagrange polyno-mials are used to interpolate the in- and out-of-surface variations of the primary variablesof each individual layer, respectively, as well as the related order of each primary variablecan be freely chosen, such as the layerwise linear, quadratic or cubic function distribution

through the thickness coordinate. The accuracy and convergence of the RMVT- and PVD-based FCLMs developed in this article are assessed by comparing their solutions with theexact 3D solutions available in the literature.

2013 Elsevier Inc. All rights reserved.

1. Introduction

In recent decades, ber-reinforced composite materials (FRCMs) have been used to form beam-, plate-, and shell-likestructures in a wide range of industry applications due to their high strength- and stiffness-to-weight ratios, in which thethrough-thickness distributions of material properties of FRCM structures are layerwise constants that always contributesto delamination and matrix cracking failures [14] at the interfaces between adjacent layers. Consequently, a new classof advanced materials, so-called functionally graded materials (FGMs), are now being instead of conventional FRCMs to formthese structures, in which the material properties of the resulting FGM structures are designed to vary continuously andsmoothly through the thickness coordinate. It has also been reported that FGM structures have better performance than con-ventional FRCM structures, due to the continuous distributions of in-surface stresses through the thickness coordinate, andlower transverse stresses at the interfaces between layers. A comprehensive survey of the relevant theoretical methodologiesand numerical modeling of both FGM and FRCM structures can be found in the literature [515] . Among these articles, theliterature survey carried out in the current work will focus on the related articles dealing with the buckling analysis of FGMand FRCM cylinders due to mechanical loads.

0307-904X/$ - see front matter 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.apm.2013.06.023

Corresponding author. Tel.: +886 6 2757575x63124; fax: +886 6 2370804.E-mail address: [email protected] (C.-P. Wu).

Applied Mathematical Modelling 38 (2014) 233252

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Applied Mathematical Modelling

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Some two-dimensional (2D) buckling analyses of laminated composite cylinders and FGM ones have been presented.Based on a higher-order theory, Simitses and Anastasiadis [16] , Anastasiadis and Simitses [17] and Anastasiadis et al. [18]studied the buckling (or instability) of moderately thick, laminated composite cylinders subjected to axial compression,external pressure, and combinations of these two loads, respectively, in which some crucial effects on the critical loads of the cylinders were examined, such as the stacking sequence, radius-to-thickness ratio, and length-to-radius ratio. Basedon the modied Donnell thin shell theory, Soyev et al. [19] and Soyev [20] investigated the elastic buckling of FGM trun-cated conical shells subjected to hydrostatic pressure and combined axial tension and hydrostatic pressure, in which thethrough-thickness distributions of the material properties of the cone were designed to have some specic functions bychanging the volume fractions of the constituents, and the material-property gradient index is concluded to have a signif-icant effect on the critical loads of the cone. This approach was also extended by Soyev [21] to the buckling analysis of FGM circular truncated conical and cylindrical shells under combined axial tension and external pressure as well as restingon the Pasternak type elastic foundation, in which the appropriate formulas for the critical loads of these shells with andwithout elastic foundations were presented. The buckling and postbuckling behaviors of cylindrical shells under combinedloads were presented by Shen and Chen [22] , in which the edge effect on the critical loads was considered using the bound-ary layer theory, and the effects of initial imperfections on the interactive buckling load and postbuckling behaviors of theshell were also discussed. Based on a higher-order shear deformation theory combined with von Karman-Donnell-type kine-matic nonlinearity, Li and Shen [23] and Shen [24] presented a postbuckling analysis of laminated composite cylindricalshells under axial compression, and combined external pressure and axial compression, in which the nonlinear prebucklingdeformations and initial geometric imperfections of the shell were taken into account. Hui and Du [25] analyzed the initialpostbuckling behavior of imperfect, antisymmetric cross-ply cylindrical shells under torsion, in which the imperfect shapewas rst considered in the same form of the torsional buckling mode. Based on a higher-order shear deformation theory [26] ,Soldatos [27,28] studied the buckling of cross-ply circular and oval cylinders, as well as antisymmetric angle-ply laminatedcylindrical panels, under axial compression. On the basis of various equivalent single-layered theories (ESLTs), a comprehen-sive survey of the buckling of laminated composite cylinders under different kinds of applied loads was carried out by Sim-itses [29] , and Anastasiadis et al. [30] .

Some exact and approximate three-dimensional (3D) buckling analyses of FGM cylinders and laminated composite oneshave been presented. Based on the 3D theory of elasticity, Kardomateas [31,32] and Kim et al. [33] presented a series of buck-ling analyses of thick orthotropic circular hollow cylinders under axial compression, external pressure, and torsion, in whichthe bifurcation of the equilibrium of these cylinders was studied, and the critical loads over a wide range of the length-to-radius and thickness-to-radius ratios were discussed. Subsequently, kardomateas [34] assessed the performance of the clas-sical and Timoshenko shell theories with regard to the buckling analysis of these cylinders under axial compression, externalpressure, and a combination of these loads, using various benchmark 3D elasticity solutions given in the above-mentionedarticles. Using a state space method combined with the successive approximation (SA) approach, Soldatos and Ye [35] and Yeand Soldatos [36] studied the 3D buckling of homogeneous/laminated composite cylinders and cylindrical panels, and thisapproach was also used for the 3D buckling analyses of thick laminated composite plates by Fan and Ye [37] and angle-plymultilayered long hollow cylinders by Ye [38] .

Within the 3D elasticity framework, Wu and Chen [39] presented asymptotic solutions for the compressive buckling of multilayered anisotropic conical shells, in which the 3D buckling problem was separated into a series of 2D buckling prob-lems governed by the partially differential equations of classical conical shell theory, and the differential quadrature (DQ)method was used to determine the critical load of each order problem. Subsequently, this 3D asymptotic approach was usedto examine both the thermoelastic buckling and thermally induced dynamic instability of laminated composite conical shellsby Wu and Chiu [40,41] . On the basis of the mixed discrete theory with a variable kinematics model, Carrera and Soave [42]and Ottavio and Carrera [43] developed an approximate 3D formulation for the linear buckling analyses of FGM structuresand laminated composite ones.

Based on the principle of virtual displacements (PVD), Cheung and Jiang [44] developed a nite rectangular layer method(FRLM) for the 3D static analysis of simply-supported, piezoelectric composite laminates, in which this semi-analytical FRLMwas demonstrated to be more effective in reducing the computational and core requirements for simply supported lami-nates. This FRLM was also extended to the 3D static, vibration, stability and thermal buckling analyses of piezoelectric com-posite plates by Akhras and Li [4547] . Subsequently, based on the Reissner mixed variational theorem (RMVT) instead of thePVD, Wu and Li [48,49] , Wu and Kuo [50] and Wu and Chang [51] developed the unied formulations of the RMVT-basedFRLMs and the nite cylindrical layer methods (FCLMs) for the 3D bending and free vibration analyses of multilayeredFGM/FRCM plates and cylinders, in which the material properties of each individual FGM layer are assumed to obey eitheran exponent-law exponentially varied with the thickness coordinate or a power-law distribution of the volume fractions of the constituents, and that the relative orders used for expansion of the displacement and transverse stress componentsthrough the thickness coordinate can be freely chosen.

After a close survey of the literature, summarized above, we found that there are relatively few articles dealing with the3D buckling analysis of multilayered FGM cylinders under axial compression, in comparison to those dealing with the 2D and3D buckling analyses of multilayered composite structures. This article thus aims at developing the unied formulations of both RMVT- and PVD-based FCLMs for the 3D buckling analysis of axially loaded, multilayered FGM and FRCM cylinders. The

linear buckling theory is used in these formulations, in which an ideal bifurcation phenomenon is assumed, and thus a mem-brane state of stresses exists just before instability occurs. Based on the RMVT and PVD, the weak formulations of the buckled

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state perturbed from the neutral equilibrium are derived. The primary variables, the displacement components and thetransverse stress ones in the RMVT as well as the displacement components only in the PVD, are expanded as the doubleFourier series in the axial and circumferential coordinates, and interpolated with the Lagrange polynomials in the thicknesscoordinate with freely chosen orders. Substituting the above-mentioned kinematic and kinetic models into the weak formu-lations, and performing the numerical integrations in the thickness coordinate, this 3D buckling problem becomes an eigen-valued one, the eigen-values and vectors of which represent the critical loads and their corresponding buckling modes,respectively. In the illustrative examples, the accuracy and convergence of these FCLMs with various orders are examinedby comparing their solutions with the exact 3D ones of multilayered composite cylinders available in the literature, and aparametric study of the inuence of some geometric and material parameters on the critical loads of multilayered FGM cyl-inders is undertaken, such as the material-property gradient index, thickness ratio for each layer, thickness-to-radius ratio,and length-to-radius ratio.

2. Pre-buckling state in a multilayered FGM cylinder

We consider a simply supported, multilayered functionally graded (FG) orthotropic circular hollow cylinder under axialcompression ( P x), as shown in Fig. 1. A global cylindrical coordinate system ( x; h and r coordinates) is adopted and located atthe center of the cylinder, and a global thickness coordinate ( f ) and a set of local thickness coordinates, z m m 1; 2; . . . ;N l,are located at the middle surfaces of the cylinder and each individual layer, respectively, as shown in Fig. 1, where N l denotesthe total number of layers constituting the cylinder. The thicknesses of each individual layer and the cylinder arehm m 1; 2; . . . ; N l and h , respectively, and h

PN lm1 hm , while R and L denote the mid-surface radius and the length of

the cylinder. The relationship between the radial coordinate and the global thickness coordinate is r R f , and that be-tween the global and local thickness coordinates in the mth-layer is f f m z m , in which f m f m fm 1 =2, and fm andf m 1 are the global thickness coordinates measured from the middle surface of the cylinder to the top and bottom surfacesof the m th-layer, respectively.

According to the assumptions of the linear instability approach, Leissa [52] notes that a set of membrane state of stressexists in the cylinder just before instability occurs. In a symmetrically FG orthotropic cylinder subjected to axial compres-sion, the displacement components of the mth-layer at the initial position are obtained by modifying the Soldatos andYes equations [35] , and are given by

um x A0 x; umh 0 ; and u

mr A0 W

m0 f m 1 ; 2 ; . . . ; N l; 1a-c

where A0 is an arbitrary constant, that will be determined later in this article by means of satisfying the force equilibriumequation in the axial direction at edges; the pre-buckling deformations in the cylinder are assumed to be axisymmetric and

plane strain.

Fig. 1. The conguration and coordinates of an FGM sandwich cylinder.

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According to the initial displacement model given in Eq. (1) , it is assumed that in the pre-buckling state the cylinder isfree of initial shear stresses (i.e., s m xr smhr s

m xh 0, m 1; 2; . . . ; N l), and the initial normal stresses in the mth-layer

can be expressed as

r m x f A0 rm x0 f ; r

mh f A0 r

mh0 f ; and r mr f A0 r

mr 0 f m 1 ; 2 ; . . . ; N l; 2a-c

where r m x0 Q m11 Q

m12 =r W

m0 Q

m13 r mr 0 ;

r mh0 Q m12 Q

m22 =r W

m0 Q

m23 r mr 0 ;

r mr 0 c m13 c

m23 =r W

m0 c

m33 W

m0 ;f ;

Q mij c mij c

mi3 c

m j3 =c

m33 i; j 1 ; 2 and 6 ; Q

mk3 c

mk3 =c

m33 k 1 and 2 ;

and c mij denotes the material elastic coefcients of the mth-layer, which is a constant for the multilayered composite cylin-der and a function of the thickness coordinate for the multilayered FGM one, while the comma denotes partial differentiationwith respect to the sufx variable.

According to the initial displacement model given in Eq. (1) , the stress equilibrium equations in the axial and circumfer-ential directions are automatically satised, and the one in the radial (or thickness) direction is given as follows:

r mr 0 ;f Q m23 1 =r r mr 0 Q m22 =r 2 W m0 Q m12 =r : 3 Using Eqs. (2c) and (3) , we can write the state space equations of the pre-buckling state of the cylinder in the following

form

dFm

df KmFm Km p ; 4

where Fm W m 0 f

r mr 0 f ( ), Km km11 km12km21 km22" #, Km p Q m13Q m12 =r ( ),km11 Q

m23 =r ; k

m12 1 =c

m33 ; k

m21 Q

m22 =r

2 and km22 Q m23 1 =r :

By means of the traction conditions imposed on the lateral surfaces of the cylinder, we can readily solve Eq. (4) using thetransfer matrix method combined with the SA method, the detailed description of which was rst given by Soldatos and

Hadjigeorgiou [53] and also can be found in Wu et al. [54,55] .In the cases of pure axial compression, the traction conditions on the lateral surfaces are

r N lr f h=2 0 and r 1

r f h=2 0 : 5

As mentioned above, the functions of W m0 f and rm r 0 f can be determined using the transfer matrix method combined

with the SA method [53] .Taking a free body diagram at each edge, we can express the force equilibrium equation in the axial direction as follows:

Z h2 ph0 Z fN lf 0 r xfrd fdh P x: 6 By satisfying Eq. (6) , we subsequently obtain the expression of A0 as follows:

A0 S xP x; 7

in which S x 2p RPN lm1 R

f mf m 1

Q m11 Q m 12 =r W

m0 Q

m13 r

mr 0 1 f =Rdfn o

1 .As a result, the initial in-surface and transverse normal stresses can be obtained as follows:

r m x f f m x fP x; rmh f f

mh fP x; and r mr f f mr fP x; 8

in which f m x , f m

h , and f mr denote the inuence functions of the initial in-surface and transverse normal stresses for the mth-

layer of the cylinder in the cases of pure axial compression, as well as f m x S xQ m

11 Q m12 =r W

m0 Q

m 13 r

mr 0 ,

f mh S xQ m12 Q

m22 =r W

m0 Q

m23 r

mr 0 , and f

mr S x r mr 0 .

3. Perturbed state in a multilayered FGM cylinder

As noted above, we aim at developing the RMVT- and PVD-based FCLMs for the 3D linear buckling analysis of simply sup-

ported, multilayered FGM and FRCM circular hollow cylinders, subjected to pure axial compression (P

x), and the detailed der-ivations of these formulations are described as follows.

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3.1. The RMVT-based FCLMs

3.1.1. Kinematic and kinetic assumptionsIn view of the use of 3D linear buckling theory, a set of in-surface and transverse normal stresses given in Eq. (8) is as-

sumed to exist in the cylinder just before instability occurs, and this is regarded as the initial state of stresses and is intro-duced in Reissners energy functional of the multilayered FGM cylinder, in which the incremental stresses associated withthe small incremental displacements perturbed from the state of neutral equilibrium will be considered.

A discrete layer model with linear, quadratic or cubic function distributions through the thickness coordinate for theincremental displacement components is adopted as the kinematic eld of the m th-layer of the cylinder in this formulation,of which the domain is in 0 6 x 6 L, 0 6 h 6 2p and hm =2 6 z m 6 hm =2, and is given by

um x x; h; z m Xnu 1

i1wmu z m iu

m x; h i; 9

umh x; h; z m Xnu 1

i1wmu z m iv

m x; h i; 10

umr x; h; z m Xnw 1

j1wmw z m j w

m x; h j; 11 where ( um x ; umh ; umr ) denote the incremental displacement components of the mth-layer of the cylinder in the x, h and r directions, respectively; umi, v mi, wm j with i 1; 2 ; . . . ;nu 1 and j 1; 2; . . . ;nw 1 are the incremental displace-

ment components at the nodal surfaces of the mth-layer of the cylinder; and wmu i i 1; . . . ;nu 1 andwmw j j 1; 2; . . . ;nw 1 are the corresponding shape functions, in which nu and nw denote the related orders used forthe expansion of the in- and out-of-surface incremental displacements, respectively.

The incremental transverse shear and normal stresses are also regarded as the primary variables in these RMVT-basedFCLMs, and are assumed as follows:

sm xr x; h; z m Xns 1

i1wms z m is

m13 x; h i; 12

sm

hr x; h; z m Xns 1

i1 wm

s z m ism23 x; h i;

13

r mr x; h; z m Xnr 1

j1wmr z m jr

m3 x; h j; 14

where s m13 i, sm23 i, r

m3 j with i 1; 2; . . . ;ns 1 and j 1; 2 ; . . . ;n r 1 are the incremental transverse stress compo-

nents at the nodal surfaces of the m th-layer of the cylinder; and wms i i 1; 2 ; . . . ;n s 1 and wmr j j 1; 2; . . . ;n r 1

are the corresponding shape functions in which ns and nr denote the related orders used for the expansion of the incremen-tal transverse shear and normal stresses, respectively.

The linear constitutive equations for the mth-layer of the cylinder, which are valid for the orthotropic materials, are givenby

rm x

r mhr mr smhr sm xr sm xh

8>>>>>>>>>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

c m11 c m12 c

m13 0 0 0

c m12 c m22 c

m23 0 0 0

c m13 c m23 c

m33 0 0 0

0 0 0 c m44 0 0

0 0 0 0 c m55 0

0 0 0 0 0 c m66

266666666664377777777775

em

xemhemr cmhr cm xr cm xh

8>>>>>>>>>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;; 15

where ( r m x ;r mh ; . . . ;sm xh ) are the incremental stress components; ( e

m x ;emh ; . . . ;c

m xh ) are the incremental strain components;

and c mij are the elastic coefcients, which are constants through the thickness coordinate in the homogeneous elastic layers,

and variable through the thickness coordinate in the FGM layers (i.e., c mij f ).The incremental strain components of each cylindrical layer based on the assumed incremental displacement compo-

nents given in Eqs. (9)(11) are written in the following form

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em x um x ; x Xnu 1

i1wmu iu

mi ; x; 16

emh 1 =r umh ;h 1 =r ur X

nu 1

i11 =r wmu iv

mi ;h X

nw 1

j11 =r wmw jw

m j ; 17

emr umr ;r Xnw 1

j1Dwmw jwm j ; 18

cm xr um x ;r umr ; x Xnu 1

i1Dwmu iu

mi X

nw 1

j1wmw jw

m j ; x; 19

cmhr 1 =r umh u

mh ;r 1 =r u

mr ;h X

nu 1

i1 1 =r wmu iv

mi Dw

mu iv

mih i X

nw 1

j11 =r wmw jw

m j ;h ; 20

cm xh 1 =r um x ;h umh ; x

X

nu 1

i11 =r wmu i u

mi ;h

X

nu 1

i1wmu iv

mi ; x; 21

where Dwmk d wmk

dz m ; and k u; w; s and r .

3.1.2. Reissners mixed variational theoremThe Reissner mixed variational theorem is used to derive the equilibrium equations at the perturbed state of the cylinder,

and its corresponding energy functional perturbed from the neutral equilibrium state is written in the form of

P R XN l

m1 Z hm =2hm =2 Z Z X r m x em x r mh emh r mr emr sm xr cm xr smhr cmhr sm xh cm xh Br mij r dxdh dz mX

N l

m1 Z hm =2hm =2 Z Z X P x f m x em x f mh emh f mr emr ; rdxd h d z m XN l

m1 Z hm =2hm =2 Z C r t mk umk dC dz m

XN l

m1

Z hm =2

hm =2

Z C u

umk umk t

mk dC dz m ; 22

where X denotes the cylinder domain on the x h surface; C r and C u denote the portions of the edge boundaries, where thesurface traction and displacement components (i.e., t mk and u

mk in which k = x, h and r ) are prescribed, respectively; Br

mij is

the complementary density function; and em x , emh and emr denote the second-order term of the incremental quantities of

GreenLagrange in-surface and transverse normal strains, respectively, and these are given by

em x um x ; x2

umh ; x2

umr ; x2h i.2 ;

emh um x ;h=r 2

umh ;h=r ur =r h i2 umr ;h=r uh=r 2 2 ;emr um x ;r

2 umh ;r

2

umr ;r 2

h i.2 :

In this RMVT-based formulation, we take both the incremental displacement and the incremental transverse stress com-ponents to be the primary variables subject to variation. Using the kinematic and kinetic assumptions, which are given inEqs. (9)(14) , respectively, we express the rst-order variation of the Reissner energy functional as follows:

dP R XN l

m1 Z hm =2hm =2 Z Z X de m p T r m p de ms T r ms der r mr dr ms T e ms Smr ms dr mr emr c

m33

1

r mr Q mr

T e m ph iordxdh dz m

XN l

m1 Z hm =2hm =2 Z Z X P x f m x dem x f mh demh f mr demr rdxd h dz m

XN l

m1 Z hm =2

hm =2

Z C r

t mk

dumk

dC dz m

XN l

m1 Z hm =2

hm =2

Z C u

umk

umk

dt mk

h idC dz m ; 23

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where the superscript of T denotes the transposition of the matrices or vectors; and C u and C r stand for the boundary edges,in which the essential and natural conditions are prescribed

e m p em x emh cm xhh iT Bm1 um Bm2 w m; e ms cm xr cmhr h iT Bm3 um Bm4 wm;

emr Bm6 wm; r m p r m x r mh s

m xhh iT Q m p Bm1 um Bm2 w m Q mr Bm7 r m;

r ms sm xr smhr h iT

Bm5 s m; r mr Bm7 r

m;

em x 1

2 um

T Bm8

T Bm8 u m

1

2 wm

T Bm9

T Bm9 w m;

emh 1

2 um

T Bm10

T Bm10 u m u m

T Bm11

T Bm12 w m

1

2 wm

T Bm13

T Bm13 w m u m

T Bm14

T Bm15 w m;

emr 1

2 um

T Bm16

T Bm16 u m

1

2 wm

T Bm6

T Bm6 w m;

u m umiv mi" #i1 ;2 ;... ;nu 1

; w m wmi

i1 ; 2 ;... ; nw 1

; s m sm13 ism23 i" #i1 ; 2 ;... ; ns 1

; r m r

m

3

i i1 ; 2 ;... ; nr 1;

Sm 1 =c m55 0

0 1 =c m44 " #; Q m p Q m11 Q

m12 0

Q m12 Q m22 0

0 0 Q m66

26643775

; Q mr Q m13Q m23

0

264 375;Q mij c

mij c

mi3 c

m j3 =c

m33 i; j 1 ; 2 ; 6 ; Q

mk3 c

mk3 =c

m33 k 1 ; 2 ;

the detailed expressions of matrices Bmk are given in Appendix A .

3.1.3. EulerLagrange equationsThe edge boundary conditions of each individual layer are considered as fully simple supports, which requires that the

following quantities are satised

umh umr r m x 0 at x 0 ; x L and m 1 ; 2 ; . . . ; N l: 24

By means of the separation of variables, the primary eld variables of each individual layer are expanded as the followingforms of a double Fourier series, so that the boundary conditions of the simply supported edges are exactly satised. Theseare given as

um x ;sm xr X1

m0X1

n0ummn ;s

m13 mn cos emx cos nh; 25

umh ;smhr X

1

m0X1

n0

v mmn ;s

m23 mn sin

emx sin nh; 26

umr ;r mr X1

m0X1

n0wmmn ;r

m3 mn sin emx cos nh: 27

where em mp=L, and m and n are positive integers and zeroes.After introducing Eqs. (25)(27) in Eq. (23) and imposing the stationary principle of the Reissner energy functional (i.e.,dP R 0), we obtain the EulerLagrange equations at the perturbed state of the cylinder as follows:

XN l

m1

KmI I KmI II K

mI III K

mI IV

KmII I KmII II K

mII III K

mII IV

KmIII I KmIII II K

mIII III 0

KmIV I KmIV II 0 K

mIV IV

26666643777775

P x

G mI I G mI II 0 0

G mII I G mII II 0 0

0 0 0 00 0 0 0

266664377775

0BBBBB@1CCCCCA

eum

e wm

esm

er m

2666437775

0000

2666437775

; 28

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where Kmi j Km j i

T i; j I; II; III; IV ; KmI I Z hm =2hm =2 eB

m1

T Q m p eB

m1 r dz m ;

KmI II Z hm =2hm =2 eBm1

T Q pB

m2 rdz m ; K

mI III Z hm =2hm =2 Bm3 T Bm5 r dz m ;

KmI IV Z

hm =2

hm =2 eBm1

T

Q mr B

m7 r dz m ; K

mII II Z

hm =2

hm =2 Bm2

T

Q m p B

m2 r dz m ;

KmII III Z hm =2hm =2 eBm4

T Bm5 r dz m ; K

mII IV Z hm =2hm =2 Bm2 T Q mr Bm7 Bm6 T Bm7h ir dz m ;

KmIII III Z hm =2hm =2 Bm5 T SmBm5 r dz m ; KmIV IV Z hm =2hm =2 1 =c m33 Bm7 T Bm7 r dz m ;G mI I Z hm =2hm =2 eB

m8

T

eBm8 r f m x eB

m10 T eB

m10 r f

mh B

m16

T Bm16 r f mr dz m ;

G mI II Z

hm =2

hm =2 eBm11

T

Bm12 r f

mh B

m14

T

eBm15 r f

mh

h idz m ;

G mII II Z hm =2hm =2 eBm9

T

eBm9 rf

m x eB

m13

T ~Bm13 rf mh B

m6

T Bm6 rf

mr h idz m ;

eum

ummn iv

mmn i264 375i1 ; 2 ;... ; nu 1 ; e

w m wmmn ih ii1 ;2 ;... ;nw 1 ;esm

sm13 mn ism23 mn i264 375i1 ; 2 ; ... ; ns 1 ;

erm r m3m n i i1 ; 2 ;... ; n r 1 ; the detailed expressions of matrices eB

m k are given in Appendix A.

After using Eq. (28) and assembling the local linear and geometric stiffness matrices of each layer constituting thecylinder by following the standard process of conventional mixed nite element methods, in which the displacement andtransverse stress continuity conditions at the interfaces between adjacent layers are imposed and thus satised a priorifor these RMVT-based FCLMs, we may construct the global linear and geometric stiffness matrices for the cylinder, whichare given as

K11 K12 K13 K14K21 K22 K23 K24K31 K32 K33 0K41 K42 0 K44

2666437775

P x

G 11 G 12 0 0G 21 G 22 0 00 0 0 00 0 0 0

2666437775

8>>>>>:9>>>=>>>;

u w

s

r

2666437775

0000

2666437775

: 29

From the last two equations in Eq. (29) , we obtain

s

r

K33 0

0 K44 1 K31 K32

K41 K42 u

w : 30

Using Eq. (30) , Eq. (29) can be condensed in the following form of

K11 P x G 11 K12 P x G 12K21 P x G 21 K22 P x G 22 K13 K14K23 K24 K33 00 K44 1 K31 K32K41 K42 ( ) u w 00 : 31

Eq. (31) represents a standard eigenvalue problem, and a nontrivial solution of this exists if the determinant of the coef-cient matrix vanishes. The critical loads, P xcr , of the cylinder for a set of xed values m ; n can be obtained by

K11 P x G 11 K12 P x G 12K21 P x G 21 K22 P x G 22 K13 K14K23 K24 K33 00 K44 1 K31 K32K41 K42 0 : 32

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Once Eq. (32) is solved, the eigenvalues and their corresponding eigenvectors, which are the critical loads and their cor-responding modal displacements at each nodal surface, respectively, can be obtained. Using this unied formulation of RMVT-based FCLMs, we may analyze the 3D linear buckling of axially loaded, multilayered FGM and FRCM cylinders, andthe performances of various RMVT-based FCLMs with different orders used for the expansion of the in- and out-of-surfacedisplacements as well as transverse shear and normal stresses can also be studied.

3.2. The PVD-based FCLMs

3.2.1. The principle of virtual displacementsThe principle of virtual displacements is a displacement-based energy principle, in which only the displacement compo-

nents are regarded as the primary variables, as given in Eqs. (9)(11) , and this principle is used to derive the equilibriumequations at the perturbed state of the cylinder, and its corresponding potential energy functional ( P p) perturbed fromthe neutral equilibrium state is written in the form of

P R XN l

m1 Z hm =2hm =2 Z Z X 1 =2 r m x em x r mh emh r mr emr sm xr cm xr smhr cmhr sm xh cm xhh irdxd h dz m XN l

m1 Z hm =2hm =2Z Z X P x f m x em x f mh emh f mr emr h i; rdxd h dz m X

N l

m1 Z hm =2hm =2 Z C r t mk umk dC dz m ; 33 In this PVD-based formulation, we take the incremental displacement components to be the primary variables subject to

variation. Using the kinematic assumptions, which are given in Eqs. (9)(11) , we express the rst-order variation of the po-tential energy functional as follows:

dP p XN l

m1 Z hm =2hm =2 ZZ X de m p T r m p de ms T r ms der r mr n ordxdh dz mX

N l

m1 Z hm =2hm =2 ZZ X P x f m x dem x f mh demh f mr demr h irdxd h d z m XN l

m1 Z hm =2hm =2 Z C r t mk dumk dC dz m ; 34 where e m p , e

ms and emr are in the same forms as those give in Eq. (23) ; and

r m p Cm p B

m1 um B

m2 w m C

mr B

m6 w m;

r ms Cms B

m3 um B

m4 w m; r mr C

mr

T Bm1 u m B

m2 w m c

m33 B

m6 wm;

Cm p

c m11 c m12 0

c m12 c m22 0

0 0 c m66

26643775

; Cms c m55 0

0 c m44" #; Cmr c m13c m23

0

264 375: 3.2.2. EulerLagrange equations

Introducing Eqs. (25)(27) in Eq. (34) and imposing the stationary principle of the potential energy functional (i.e.,dP p 0), we obtain the EulerLagrange equations at the perturbed state of the cylinder as follows:

XN l

m1

DmI I DmI II

DmII I DmII II

" # P x G mI I G mI IIG mII I G mII II" # ! e

u m

e w m" # 00 ; 35

where

Dmi j Dm j i

T i; j I; II; DmI I Z hm =2hm =2 eB

m1

T Cm p eB

m1 B

m3

T Cms B

m3h ir dz m ;

DmI II Z hm =2hm =2 eBm1

T C p B

m2 C

mr B

m6 B

m3

T Cms eB

m4h ir dz m ;

DmII II Z hm =2hm =2 Bm2 T Cm p Bm2 Cmr Bm6 eBm4

T Cms eB

m4 B

m6

T Cmr

T Bm2 B

m6

T c m33 B

m6n or dz m :

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After using Eq. (35) and assembling the local linear and geometric stiffness matrices of each layer constituting the cylin-der by following the standard process of conventional displacement-based nite element methods, in which the displace-ment continuity conditions at the interfaces between adjacent layers are imposed and thus satised a priori for thesePVD-based FCLMs, we may construct the global linear and geometric stiffness matrices for the cylinder, which are given as

D11 D12D21 D22 P x G 11 G 12G 21 G 22 u w 00 : 36

Eq. (36) represents a standard eigenvalue problem, and a nontrivial solution of this exists if the determinant of the coefcientmatrix vanishes. The critical loads of the cylinder for a set of xed values m ; n can be obtained by

D11 D12D21 D22 P x G 11 G 12G 21 G 22 0 : 37

Once Eq. (37) is solved, the critical loads and their corresponding modal displacements at each nodal surface can be ob-tained. Using the unied formulation of PVD-based FCLMs, we may analyze the 3D linear buckling of axially loaded, multi-layered FGM and FRCM cylinders, and the performances of various PVD-based FCLMs with different orders used for theexpansion of the in- and out-of-surface displacements can also be studied.

4. Illustrative examples

In the following examples, LM ns nrnu nw is dened to represent the RMVT-based FCLMs, in which the in- and out-of-surfacedisplacements are expanded as the nu and nw -order Lagrange polynomials, respectively, while the transverse shear andnormal stresses are expanded as the n s - and nr -order Lagrange polynomials in the thickness coordinate of each layer, respec-tively; and LD nu nw is dened to represent the PVD-based FCLMs, in which the in- and out-of-surface displacements are ex-panded as the nu - and nw -order Lagrange polynomials, respectively, in the thickness coordinate of each layer. In addition, anh-renement process is adopted for this work to obtain the convergent solutions of these RMVT- and PVD-based FCLMs, andthe values of n i i u; v ; s and r are taken as 1, 2 or 3.

Table 1

The present solutions of various PVD- and RMVT-based FCLMs for the critical load parameters of laminated [0 /90 ]20 orthotropic cylinders under axialcompression and with different buckling modes ( N l = 40, L/R = 5, h/R = 0.2, P xcr P xcr R

2 =2 p RE T h3 ).

m ; n Theories

LD11 LD22 LD33 LM1111 LM2222 LM

3333

Noor [56] Ye and Soldatos [36]

(1, 0) 12.554 12.555 12.555 12.555 12.555 12.555 12.50 12.520(1, 1) 5.509 5.509 5.509 5.509 5.509 5.509 5.511 5.52(1, 2) 11.039 11.037 11.037 11.037 11.037 11.037 11.08 11.104(1, 3) 57.069 57.050 57.050 57.050 57.050 57.050 57.70 57.584(1, 4) 154.502 154.427 154.427 154.453 154.427 154.427 159.4 NA(1, 5) 296.545 296.357 296.357 296.378 296.357 296.357 319.1 NA(2, 0) 12.555 12.555 12.555 12.555 12.555 12.555 12.50 12.564(2, 1) 6.378 6.378 6.378 6.378 6.378 6.378 6.380 6.391(2, 2) 5.451 5.450 5.450 5.450 5.450 5.450 5.466 5.467(2, 3) 16.458 16.453 16.453 16.455 16.453 16.453 16.54 16.674(2, 4) 41.456 41.437 41.437 41.441 41.437 41.437 41.83 NA(2, 5) 80.418 80.373 80.373 80.389 80.373 80.373 81.77 NA(3, 0) 12.555 12.555 12.555 12.555 12.555 12.555 12.50 12.572(3, 1) 6.834 6.834 6.834 6.834 6.834 6.834 6.838 6.849(3, 2) 5.038 5.038 5.038 5.038 5.038 5.038 5.052 5.051(3, 3) 9.587 9.584 9.584 9.584 9.584 9.584 9.622 9.613(3, 4) 20.675 20.667 20.666 20.671 20.666 20.666 20.79 NA(3, 5) 38.218 38.198 38.197 38.203 38.197 38.197 38.54 NA(4, 0) 12.555 12.555 12.555 12.555 12.555 12.555 NA NA(4, 1) 7.234 7.234 7.234 7.234 7.234 7.234 NA NA(4, 2) 5.361 5.361 5.361 5.361 5.361 5.361 NA NA(4, 3) 7.691 7.689 7.689 7.689 7.689 7.689 NA NA(4, 4) 13.881 13.875 13.875 13.876 13.875 13.875 NA NA(4, 5) 23.783 23.771 23.771 23.778 23.771 23.771 NA NA(5, 0) 12.555 12.555 12.555 12.555 12.555 12.555 NA NA(5, 1) 7.595 7.594 7.594 7.594 7.594 7.594 NA NA(5, 2) 5.841 5.840 5.840 5.840 5.840 5.840 NA NA(5, 3) 7.172 7.170 7.170 7.170 7.170 7.170 NA NA(5, 4) 11.089 11.085 11.085 11.086 11.085 11.085 NA NA

(5, 5) 17.427 17.419 17.419 17.419 17.419 17.419 NA NA

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4.1. Multilayered composite material cylinders

The benchmark solutions of the critical loads of simply supported, 0 =90 20 laminated circular hollow cylinders subjected to axial compression were presented by Noor and Peters [56] and Ye and Soldatos [36] , and they are used to validatethe accuracy and convergence of the present solutions obtained using the above-mentioned RMVT- and PVD-based FCLMs inTable 1 , in which the bers of the different layers alternate between the circumferential and longitudinal directions, with thebers of the top and bottom layers running in the circumferential and longitudinal directions, respectively, the geometricparameters of the cylinders are L=R 5 and h=R = 0.2, and the material properties of the cylinders are given as

E L=E T 15 ; GLT =E T 0 :5 ; GTT =E T 0 :35 and t LL t LT 0 :3 ; 38a-d

where the subscripts of L and T denote the directions parallel and transverse to the ber directions, respectively.For compar-ison purposes, the critical load parameter, P xcr , is dened as

P xcr P xcr R2 =2 pRE T h

3: 39

Table 1 shows the present solutions of various PVD- and RMVT-based FCLMs for the critical load parameters of 0 =90 20laminated circular hollow cylinders under axial compression and with different buckling modes, m 1-5 and n 0-5. Itcan be seen from Table 1 that the forty-layer solutions of LD 22 , LD33 , LM2222 , and LM

3333 are almost identical to one another,

and are in excellent agreement with the exact 3D solutions available in the literature. It is also shown that for a xed valueof m the critical load parameter rst monotonically decrease as the value of n rises, and then it increases, while for a xedvalue of n there is no consistent tendency for the variation of the critical load parameter with the increasing value of m , and

the lowest critical load parameter occurs at the buckling mode, m ; n 3; 2.Table 2 shows the convergence studies for the present solutions of the lowest critical load parameters of 0 =90 S lam-inated hollow cylinders under axial compression and with different orthotropic ratios ( E L=E T ) and different total number of layers ( N l), in which L/R = 5, h/R = 0.2; E L=E T = 5, 10, 20, 30 and 40, and other material properties are identical to those used inTable 1 ; N l = 4, 8, 16, 32 and 40. It can be seen in Table 2 that the convergent solutions of various PVD- and RMVT-solutions

Table 2

Convergence studies for the present solutions of the lowest critical load parameters of laminated 0 =90 s cylinders under axial compression and with differentorthotropic ratios ( L/R = 5, h/R = 0.2, and P xcr P xcr R

2 =2 p RE T h3 ).

Theories E L/E T ; m ; n

5; (2,2) 10; (2, 2) 20; (2, 2) 30; (2, 2) 40; (1, 1)

Present LD 11 (N l = 4) 3.1614 (3.1635) 3.9945 (3.9967) 5.1512 (5.1532) 6.0298 (6.0318) 6.4475 (6.4481)

Present LD 11 (N l = 8) 3.1519 (3.1540) 3.9791 (3.9813) 5.1184 (5.1205) 5.9751 (5.9770) 6.4440 (6.4447)Present LD 11 (N l = 16) 3.1494 (3.1515) 3.9751 (3.9772) 5.1093 (5.1113) 5.9596 (5.9615) 6.4431 (6.4438)Present LD 11 (N l = 32) 3.1488 (3.1508) 3.9740 (3.9761) 5.1069 (5.1090) 5.9557 (5.9576) 6.4429 (6.4436)Present LD 11v (N l = 40) 3.1487 (3.1507) 3.9739 (3.9760) 5.1067 (5.1087) 5.9552 (5.9571) 6.4429 (6.4436)Present LD 22 (N l = 4) 3.1483 (3.1504) 3.9735 (3.9757) 5.1069 (5.1090) 5.9564 (5.9583) 6.4428 (6.4435)Present LD 22 (N l = 8) 3.1485 (3.1505) 3.9735 (3.9757) 5.1061 (5.1081) 5.9543 (5.9562) 6.4428 (6.4435)Present LD 22 (N l = 16) 3.1486 (3.1506) 3.9736 (3.9758) 5.1061 (5.1081) 5.9543 (5.9562) 6.4428 (6.4435)Present LD 22 (N l = 32) 3.1486 (3.1506) 3.9736 (3.9758) 5.1061 (5.1082) 5.9543 (5.9562) 6.4428 (6.4435)Present LD 22 (N l = 40) 3.1486 (3.1506) 3.9737 (3.9758) 5.1061 (5.1082) 5.9543 (5.9562) 6.4428 (6.4435)Present LD 33 (N l = 4) 3.1481 (3.1502) 3.9731 (3.9753) 5.1054 (5.1075) 5.9535 (5.9554) 6.4428 (6.4435)Present LD 33 (N l = 8) 3.1485 (3.1505) 3.9735 (3.9757) 5.1060 (5.1080) 5.9541 (5.9560) 6.4428 (6.4435)Present LD 33 (N l = 16) 3.1486 (3.1506) 3.9736 (3.9758) 5.1061 (5.1081) 5.9543 (5.9562) 6.4428 (6.4435)Present LD 33 (N l = 32) 3.1486 (3.1506) 3.9736 (3.9758) 5.1061 (5.1082) 5.9543 (5.9562) 6.4428 (6.4435)Present LD 33 (N l = 40) 3.1486 (3.1506) 3.9737 (3.9758) 5.1061 (5.1082) 5.9543 (5.9562) 6.4428 (6.4435)Present LM 1111 (N l = 4) 3.1481 (3.1503) 3.9736 (3.9757) 5.1091 (5.1112) 5.9621 (5.9640) 6.4444 (6.4451)

Present LM 1111 (N l = 8) 3.1486 (3.1506) 3.9737 (3.9758) 5.1063 (5.1083) 5.9546 (5.9565) 6.4428 (6.4435)

Present LM 1111 (N l = 16) 3.1486 (3.1506) 3.9736 (3.9758) 5.1061 (5.1082) 5.9543 (5.9562) 6.4428 (6.4435)

Present LM 1111 (N l = 32) 3.1486 (3.1506) 3.9736 (3.9758) 5.1061 (5.1082) 5.9543 (5.9562) 6.4428 (6.4435)

Present LM 1111 (N l = 40) 3.1486 (3.1506) 3.9737 (3.9758) 5.1061 (5.1082) 5.9543 (5.9562) 6.4428 (6.4435)

Present LM 2222 (N l = 4) 3.1482 (3.1503) 3.9733 (3.9755) 5.1062 (5.1081) 5.9548 (5.9567) 6.4428 (6.4435)

Present LM 2222 (N l = 8) 3.1485 (3.1505) 3.9735 (3.9757) 5.1060 (5.1082) 5.9542 (5.9561) 6.4428 (6.4435)

Present LM 2222 (N l = 16) 3.1486 (3.1506) 3.9736 (3.9758) 5.1061 (5.1081) 5.9543 (5.9562) 6.4428 (6.4435)

Present LM 2222 (N l = 32) 3.1486 (3.1506) 3.9736 (3.9758) 5.1061 (5.1082) 5.9543 (5.9562) 6.4428 (6.4435)

Present LM 2222 (N l = 40) 3.1486 (3.1506) 3.9737 (3.9758) 5.1061 (5.1082) 5.9543 (5.9562) 6.4428 (6.4435)

Present LM 3333 (N l = 4) 3.1481 (3.1502) 3.9731 (3.9753) 5.1054 (5.1075) 5.9535 (5.9554) 6.4428 (6.4435)

Present LM 3333 (N l = 8) 3.1485 (3.1505) 3.9735 (3.9757) 5.1060 (5.1080) 5.9541 (5.9560) 6.4428 (6.4435)

Present LM 3333 (N l = 16) 3.1486 (3.1506) 3.9736 (3.9758) 5.1061 (5.1081) 5.9543 (5.9562) 6.4428 (6.4435)

Present LM 3333 (N l = 32) 3.1486 (3.1506) 3.9736 (3.9758) 5.1061 (5.1082) 5.9543 (5.9562) 6.4428 (6.4435)

Present LM 3333 (N l = 40) 3.1486 (3.1506) 3.9737 (3.9758) 5.1061 (5.1082) 5.9543 (5.9562) 6.4428 (6.4435)

The solutions in the parentheses are obtained without consideration of the effect of initial transverse normal stress.

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0 5 10 15 20

L / R

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

( P ~

x ) c r

m^ =1

m=2

m=3

m^ =4

m=5Lowest criticalload parameters

(a)

(b)

(c)

Fig. 2. Variations of the critical load parameters of axially loaded [0 /90 ]s laminated cylinders with the length-to-radius ratio for m = 15, (a) R/h = 20, (b)R/h = 10, (c) R/h = 5.


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(a)

(b)

(c)

Fig. 3. Variations of the critical load parameters of axially loaded, [0 /90 ]s laminated cylinders with the radius-to-thickness ratio for m = 15, (a) L/R = 5, (b)L/R = 10, (c) L/R = 20.


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are obtained when N l 40 for a wide range of the orthotropic ratio (i.e., E L=E T = 540). In addition, the solutions in paren-theses are obtained without consideration of the effect of initial transverse normal stress (i.e., let r mr 0), and the effectalways decreases the present solutions of various FCLMs, although the difference between them is no more than 0.06%.Therefore, the effect of the initial transverse normal stress on the critical load parameters is concluded to be very minorin the cases of buckling of axially loaded cylinders.

Figs. 2 and 3 show the variations of the forty-layered LM3333 solutions of the critical load parameter eP

xcr of axiallyloaded, 0 =90 S laminated hollow cylinders with the length-to-radius ratio and the radius-to-thickness ones, respectively,for m 1-5, in which R/h = 20, 10 and 5 in Fig. 2; L/R = 5, 10 and 20 in Fig. 3; E L=E T = 30, and other material properties areidentical to those used in Table 1 ; eP xcr P xcr =2pRE T h. Referring to the gures, the magnitude of the lowest critical loadparameter and its corresponding number of half-waves ( m ) for a wide range of the length-to-radius and radius-to-thicknessratios are shown using a solid dark line and can readily be found. It is also observed that the lowest critical load increases asthe laminated hollow cylinder becomes thicker for a specic length-to-radius ratio and as this becomes shorter for a specicradius-to-thickness ratio.

4.2. FGM sandwich cylinders

The buckling of simply-supported, axially loaded sandwich FGM cylinders consisting of a soft FGM core layer bounded

with two stiff homogeneous face sheets (i.e., homogeneous material =FGM=homogeneous material cylinders) is studied.The dimensionless critical load parameters are dened as the same forms used in the Example 4.1 except that E T is replaced

Table 3

Convergence studies for the present solutions of the critical load parameters of FGM sandwich cylinders under axialcompression and with different material-property gradient indices ( L/R = 5, h/R = 0.1, h1:h2:h3=0.1h:0.8 h:0.1 h,

^

m; ^

n=(1, 2), and P xcr P xcr R2 =2p R E f h

3 .

Theories j

0 1 5 10 1

Present LD 22 (N l = 10) 3.5853 2.8212 2.1417 1.9459 1.6858Present LD 22 (N l = 20) 3.5854 2.8295 2.1556 1.9593 1.6859Present LD 22 (N l = 40) 3.5854 2.8316 2.1593 1.9631 1.6859Present LD 22 (N l = 80) 3.5854 2.8321 2.1602 1.9641 1.6859Present LD 33 (N l = 10) 3.5853 2.8212 2.1417 1.9459 1.6858Present LD 33 (N l = 20) 3.5854 2.8295 2.1556 1.9593 1.6859Present LD 33 (N l = 40) 3.5854 2.8316 2.1593 1.9631 1.6859Present LD 33 (N l = 80) 3.5854 2.8321 2.1602 1.9641 1.6859Present LM 2222 (N l = 10) 3.5853 2.8212 2.1417 1.9459 1.6858

Present LM 2222 (N l = 20) 3.5854 2.8295 2.1556 1.9593 1.6859

Present LM 2222 (N l = 40) 3.5854 2.8316 2.1593 1.9631 1.6859

Present LM 2222 (N l = 80) 3.5854 2.8321 2.1602 1.9641 1.6859

Present LM 3333 (N l = 10) 3.5853 2.8212 2.1417 1.9459 1.6858

Present LM 3333 (N l = 20) 3.5854 2.8295 2.1556 1.9593 1.6859

Present LM 3333 (N l = 40) 3.5854 2.8316 2.1593 1.9631 1.6859

Present LM 3333 (N l = 80) 3.5854 2.8321 2.1602 1.9641 1.6859

Table 4

The present solutions of the lowest critical load parameters of FGM sandwich cylinders under axial compression ( L/R = 5, h/R = 0.2, andP xcr P xcr R

2 =2p RE f h3 ).

h1:h2:h3 m ; n Theories j

0 1 5 10 1

0.1 h:0.8 h:0.1 h (2, 2) Present LD 22 1.8415 1.3836 1.0128 0.9173 0.7901Present LD 33 1.8415 1.3836 1.0128 0.9173 0.7901Present LM 2222 1.8415 1.3837 1.0129 0.9174 0.7901

Present LM 3333 1.8415 1.3837 1.0129 0.9174 0.79010.2 h:0.6 h:0.2 h (2, 2) Present LD 22 1.8415 1.5168 1.2547 1.1904 1.1093

Present LD 33 1.8415 1.5168 1.2547 1.1904 1.1093

Present LM2222

1.8415 1.5168 1.2548 1.1904 1.1093

Present LM 3333 1.8415 1.5168 1.2548 1.1904 1.1093(h/3):( h/3):( h/3) (2, 2) Present LD 22 1.8415 1.6693 1.5257 1.4905 1.4469

Present LD 33 1.8415 1.6693 1.5257 1.4905 1.4469Present LM 2222 1.8415 1.6692 1.5256 1.4904 1.4470

Present LM 3333 1.8415 1.6692 1.5256 1.4904 1.4470

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(a)

(b)

(c)

Fig. 4. Variations of the critical load parameters of axially loaded, FGM sandwich cylinders with the length-to-radius ratio for m = 15, (a) j = 1, (b) j = 5, (c)j = 1 .


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(a)

(b)

(c)

Fig. 5. Variat ions of the critical load parameters of axially loaded, FGM sandwich cylinders with the radius-to-thickness ratio for m = 15, (a) j = 1, (b) j = 5,(c) j = 1 .


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by E f , which is the Youngs modulus of the face sheets. The thickness ratio of each layer of the sandwich cylinder ish1 : h2 : h3 , in which h1 h3 and P3m1 hm h, while the effective engineering constants of each layer are written as follows:E mf E 0 E f E 0 C mf m 1 ; 2 and 3 ; 40a

t m constant m 1 ; 2 and 3 ; 40b

where E 0 denotes the Youngs modulus of the material at the mid-surface of the core, for which E 0 = 70 GPa (Aluminum) and

E f = 380 GPa (Alumina) are used in this example; tm

(m = 13) are taken to be 0.3; Cm

(m = 13) are the volume fractions of the constituents of the cylinder, and are given by

C 1 1 when h=2 6 f 6 h2 =2 ; 41a

C 2 f jf j=h2 =2 j when h2 =2 6 f 6 h2 =2 ; 41b

C 3 1 when h2 =2 6 f 6 h=2 ; 41c

where j denotes the material-property gradient index.It is apparent that when j = 0, C 2 1, this FGM sandwich cylinder reduces to a single-layered homogeneous cylinder

with material properties E f = 380 GPa and t f = 0.3; while when j 1 , C 2 0, this FGM sandwich cylinder reduces to ahomogeneous sandwich cylinder with material properties E 1 E 3 = 380 GPa, E 2 = 70 GPa, and t m = 0.3 (m = 13).

Table 3 shows the convergence studies for the present solutions of the lowest critical load parameters of axially loaded,FGM sandwich cylinders with different values of the material-property gradient indices, in which N l = 10, 20, 40 and 80; L/R = 5, h/R = 0.1 and h1 : h2 : h3 0:1h : 0:8h : 0:1h; m ; n 1 ; 2. Again, it can be seen in Table 3 that the solutions of LD 22 ,LD33 , LM2222 and LM

3333 are identical to one another, and the convergent solutions are obtained as N l 80. The lowest critical

load parameter decreases when the material-property gradient index becomes large, which means that the cylinder becomessoft.

Table 4 shows the convergent solutions of LD 22 , LD33 , LM2222 and LM3333 for the lowest critical load parameters of FGM sand-

wich cylinders under axial compression and with different thickness ratios for each layer and different material-propertygradient index, in which L/R = 5, h /R = 0.2, and N l 80. It can be seen in Table 4 that the lowest critical load parameter in-creases when the core thickness becomes thin, which means that the cylinder becomes stiff.

Figs. 4 and 5 show the variations of the forty-layered LM 3333 solutions of the critical load parameter eP xcr of axiallyloaded, FGM sandwich cylinders with the length-to-radius ratio and the radius-to-thickness one, respectively, form 1 5, in which h/R = 0.1, j = 1; 5 and 1 in Fig. 4; L/R = 5, j = 1; 5 and 1 in Fig. 5. Again, referring to the gures,the magnitude of the lowest critical load parameter and its corresponding number of half-waves ( m ) for a wide range of the length-to-radius and radius-to-thickness ratios are shown using a solid dark line and can readily be found. It is also ob-served that the lowest critical load decreases as the material-property gradient index ( j ) becomes large, while the corre-sponding buckling modes are not affected by changing the values of j .

5. Concluding remarks

In this work, we developed a unied formulation of various PVD- and RMVT-based FCLMs to investigate the 3D linearbuckling of simply supported, FGM sandwich cylinders and laminated composite ones under pure axial compression. It isshown in the illustrative examples that the present LD 22 , LD33 , LM2222 and LM

3333 solutions of critical load parameters of lam-

inated orthotropic hollow cylinders and FGM sandwich ones are almost identical to one another, and the convergent solu-tions of these are in excellent agreement with the exact 3D solutions available in the literature. A parametric study of variouseffects on the critical load parameter of axially loaded, FGM sandwich cylinders and laminated composite ones and their cor-responding buckling modes was undertaken, examining the effects of the length-to-radius, radius-to-thickness, orthotropicratios and material-property gradient index. It is concluded that the effect of the initial transverse normal stress on the crit-ical load parameters is very minor in the cases of buckling of axially loaded cylinders, that the lowest critical load increasesas the laminated orthotropic cylinder becomes thicker for a specic length-to-radius ratio and as this becomes shorter for aspecic radius-to-thickness ratio, and that the lowest critical load decreases as the material-property gradient index ( j ) be-comes larger, while the corresponding buckling modes are not affected by changing the values of j .

As we mentioned above, the FGM sandwich cylinders are superior to the homogeneous sandwich ones because the mate-rial properties of FGM sandwich cylinder can be designed as the continuous function distributions through the thicknesscoordinate without any mismatch at the interfaces between the face sheet and core. The present PVD- and RMVT-basedFCLMs may become a powerful and efcient numerical tool for the analysis of various mechanical problems of these FGMsandwich structures, and their convergent solutions may provide a standard for assessing the accuracy and convergence rateof a variety of 2D theoretical methodologies and numerical models of FGM structures. Furthermore, these are also capable of being extensively used to the coupled analysis of some other advanced material structures, such as the carbon nanotube-

reinforced polymer structures, the functionally graded piezoelectric structures, multiwalled carbon nanotubes and multilay-ered graphene sheets, which will be continuously working on.

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Acknowledgment

This work was supported by the National Science Council of Republic of China through Grant NSC 100-2221-E-006-180-MY3.

Appendix A.

The detailed expressions of matrices Bmk and eB

m k are given by

Bm1 wmu i@ x 0

0 1 =r wmu i@ h1 =r wmu i@ h w

mu i@ x

26643775i1 ; 2 ;... ; nu 1

; Bm2

0

1 =r wmw i0

264 375i1 ; 2 ;... ; nw 1 ;Bm3

Dwmu i 0

0 1 =r wmu i Dwmu i" #i1 ; 2 ;... ; nu 1 ; Bm4 w

mw i @ x

1 =r wmw i@ h" #i1 ; 2 ;... ; nw 1 ;Bm5

wms i 0

0 wms i" #i1 ; 2 ;... ; ns 1 ; Bm6 Dwmw ih ii1 ; 2 ;... ; nw 1 ; Bm7 wmr ih ii1 ; 2 ;... ; nr 1 ;Bm8

wmu i @ x 0

0 wmu i @ x" #i1 ; 2 ;... ; nu 1 ; Bm9 wmw i @ xh ii1 ; 2 ;... ; nw 1 ;Bm10

1 =r wmu i @ h 0

0 1 =r wmu i 1 @ h" #i1 ; 2 ;... ; nu 1 ; Bm11 0 00 1 =r wmu i @ h i1 ; 2 ;... ; nu 1 ;Bm12

0

1 =r wmw i i1 ; 2 ;... ; nw 1 ; Bm13 1 =r wmw i 1 @ h i1 ; 2 ;... ; nw 1 ;Bm14

0 0

0

1=r w

m

u i i1 ; 2 ;... ; nu 1

; Bm15 0

1

=r wm

w i @ h i1 ; 2 ;... ; nw 1

;

Bm16 Dwmu i 0

0 Dwmu i" #i1 ; 2 ;... ; nu 1 ;

eBm1 em w

mu i 0

0 n 1 =r wmu in 1 =r wmu i em w

mu i

26643775i1 ; 2 ;... ; nu 1

; eBm4 em w

mw i

en 1 =r wmw i" #i1 ; 2 ;... ; nw 1 ;

eBm8

em wmu i 0

0

em wmu i" #i1 ; 2 ;... ; nu 1 ;

eBm9

em wmw i

h ii1 ; 2 ;... ; nw 1

;

eBm10

n=r wmu i 0

0 1 =r wmu i 1 n" #i1 ; 2 ;... ; nu 1 ; eBm11

0 0

0 n=r wmu i i1 ; 2 ;... ; nu 1eB

m13 1 =r w

mw i 1 nh ii1 ; 2 ;... ; nw 1 ; eB

m15

0

n=r wmw i i1 ; 2 ;... ; nw 1 :References

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